OP-01041 Partially resolved

The big-line-big-clique conjecture

Points of a set $P \subset \mathbb{R}^2$ are mutually visible if the segment joining any two of them contains no other point of $P$. The statement below was known to Green for $\ell \le 5$. A set with no $\ell$ mutually visible points has $\gg_\ell n^2$ collinear triples.

Problem. Fix integers $k$ and $\ell$. Is it true that every set of $n \ge n_0(k, \ell)$ points in $\mathbb{R}^2$ contains either $k$ points on a line or $\ell$ mutually visible points?

Source: J. Kára, A. Pór and D. R. Wood, On the chromatic number of the visibility graph of a set of points in the plane, Discrete Comput. Geom. 34 (2005). Listed as Problem 70 in B. Green, 100 open problems (version of January 2026).

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: CO — Combinatorics
Secondary: MG
Keywords: collinear points, discrete geometry, Ramsey theory, visibility graphs

References

@article{kara2005visibility,
  author = {K{\'a}ra, Jan and P{\'o}r, Attila and Wood, David R.},
  title = {On the chromatic number of the visibility graph of a set of points in the plane},
  journal = {Discrete \& Computational Geometry},
  volume = {34},
  number = {3},
  pages = {497--506},
  year = {2005},
  doi = {10.1007/s00454-005-1177-z}
}

@article{abel2011pentagon,
  author = {Abel, Zachary and Ballinger, Brad and Bose, Prosenjit and Collette, S{\'e}bastien and Dujmovi{\'c}, Vida and Hurtado, Ferran and Kominers, Scott Duke and Langerman, Stefan and P{\'o}r, Attila and Wood, David R.},
  title = {Every large point set contains many collinear points or an empty pentagon},
  journal = {Graphs and Combinatorics},
  volume = {27},
  number = {1},
  pages = {47--60},
  year = {2011},
  doi = {10.1007/s00373-010-0957-2}
}

@misc{bonnet2026clique,
  author = {Bonnet, {\'E}douard},
  title = {Large finite point sets have 4 collinear points or a 6-clique},
  year = {2026},
  eprint = {2608.19468},
  archivePrefix = {arXiv}
}

@misc{cambie2026visible,
  author = {Cambie, Stijn},
  title = {Four collinear points or six visible points},
  year = {2026},
  eprint = {2609.21035},
  archivePrefix = {arXiv}
}

@misc{bhattacharya2026almost,
  author = {Bhattacharya, Bhaswar B. and Das, Sandip and Islam, Sk Samim and Mohapatra, Aashirwad and Sen, Saumya},
  title = {Large planar point sets contain 4 collinear points or almost 7-cliques, and related results},
  year = {2026},
  eprint = {2609.25727},
  archivePrefix = {arXiv}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 70; version of January 2026, most recent update December 2025},
  year = {2026}
}

Resolution claims

Large Finite Point Sets Have 4 Collinear Points or a 6-Clique v1 arXiv ↗ AI-Verified

Édouard Bonnet

We prove that every finite point set of size at least $10^{11055931}$ has four collinear points or six points that pairwise see each other. This resolves the first open case of the big-line-big-clique conjecture of Kára, Pór, and Wood.

Partial resolution · not verified · fidelity not established · posted on arXiv 19 Aug 2026

Status history

When	Change	By	Reason
2026-10-29	Open → Partially resolved	The registry's curators	Large Finite Point Sets Have 4 Collinear Points or a 6-Clique v1 recorded from arXiv:2608.19468v1.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01041, "The big-line-big-clique conjecture", clopen.solutions.